0-1 law for percolation on sufficiently regular rhombus tilings

Establish a 0–1 law for percolation processes on sufficiently regular edge-to-edge rhombus tilings, including Penrose tilings and multigrid dual tilings, under i.i.d. Bernoulli initial configurations; specifically, prove that the probability of percolation is either 0 or 1 and never strictly between these values.

Background

On the integer lattice Z^2, percolation processes are known to obey a 0–1 law: when the initial configuration is sampled from a Bernoulli product measure, the probability that percolation occurs is either 0 or 1. This classical result is attributed to van Enter (1987) for bootstrap percolation and related processes.

The paper studies growth dynamics on multigrid dual tilings (notably including Penrose tilings) and suggests extending percolation-type analyses beyond Z^2. The authors propose that an analogous 0–1 law should hold on sufficiently regular rhombus tilings, encompassing Penrose and multigrid dual tilings, under Bernoulli initial conditions.